In this paper, we employ different techniques as compared to [1]} that generalize the classical Schensted's insertion algorithm and RSK correspondence to the complete flag variety case, using bumpless pipe dreams and decorated chains in Bruhat order. Our insertion algorithms come from understanding Monk's rule for Schubert polynomials bijectively using bumpless pipe dreams. More specifically, we give two different bijective proofs of Monk's rule that generalize the row and column insertions on semi-standard Young tableaux. The right insertion algorithm on bumpless pipe dreams is an extension of the first author's bijective proof of the one-variable version of Monk's rule [2]}, and the left insertion algorithm presented here is new. Bijective proofs of Monk's rule have been presented with pipe dreams (RC-graphs) in [3]} and [4]}. Under the bijection given in [5]}, we conjecture that our left insertion algorithm on bumpless pipe dreams corresponds to the insertion algorithm on pipe dreams for Monk's rule given in [3]}, and it was shown in [5]} that the algorithm in [2]} corresponds to the algorithm in [4]}. Therefore, our story about RSK could conceivably be told in terms of pipe dreams as well as bumpless pipe dreams. We work out the bumpless pipe dream version in full technical detail.
